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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 3.5.52
Chapter 3, Problem 3.5.52

Is there a value of b that will make


g(x) = { x + b, x < 0
cos x, x ≥ 0


continuous at x = 0? Differentiable at x = 0? Give reasons for your answers.

Verified step by step guidance
1
To determine if the function g(x) is continuous at x = 0, we need to check if the left-hand limit as x approaches 0 from the negative side equals the right-hand limit as x approaches 0 from the positive side, and both equal g(0).
Calculate the left-hand limit: As x approaches 0 from the left (x < 0), g(x) = x + b. The limit is lim(x→0⁻)(x + b) = 0 + b = b.
Calculate the right-hand limit: As x approaches 0 from the right (x ≥ 0), g(x) = cos(x). The limit is lim(x→0⁺)cos(x) = cos(0) = 1.
For g(x) to be continuous at x = 0, the left-hand limit must equal the right-hand limit and g(0). Therefore, set b = 1 to make the function continuous at x = 0.
To determine differentiability at x = 0, check if the derivative from the left equals the derivative from the right. The derivative of x + b is 1, and the derivative of cos(x) at x = 0 is 0. Since these derivatives are not equal, g(x) is not differentiable at x = 0.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Continuity

A function is continuous at a point if the limit of the function as it approaches the point from both sides equals the function's value at that point. For g(x) to be continuous at x = 0, the left-hand limit (as x approaches 0 from the left) and the right-hand limit (as x approaches 0 from the right) must both equal g(0).
Recommended video:
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Intro to Continuity

Differentiability

A function is differentiable at a point if it has a defined derivative at that point, meaning the function's rate of change is consistent from both sides. For g(x) to be differentiable at x = 0, it must first be continuous at x = 0, and the left-hand derivative and right-hand derivative at x = 0 must be equal.
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Finding Differentials

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals. Understanding how to evaluate limits and derivatives for each piece is crucial. For g(x), we must analyze the behavior of x + b for x < 0 and cos x for x ≥ 0 separately, then ensure they align at x = 0 for continuity and differentiability.
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Piecewise Functions
Related Practice
Textbook Question

In Exercises 41–44, determine whether the piecewise-defined function is differentiable at x = 0.


g(x) = { x²/³, x ≥ 0

x¹/³, x < 0

Textbook Question

Differential Estimates of Change


In Exercises 35–40, write a differential formula that estimates the given change in volume or surface area.


The change in the lateral surface area S = πr√(r² + h²) of a right circular cone when the radius changes from r₀ to r₀ + dr and the height does not change

Textbook Question

a. Graph the function


ƒ(x) = { x², -1 ≤ x < 0

{ -x², 0 ≤ x ≤ 1.


b. Is ƒ continuous at x = 0?

c. Is ƒ differentiable at x = 0?


Give reasons for your answers.

Textbook Question

Find the points on the curve y = 2x³ - 3x² - 12x + 20 where the tangent line is


a. perpendicular to the line y = 1 - (x/24).

_

b. parallel to the line y = √2 - 12x.

Textbook Question

Find the derivatives of the functions in Exercises 1–42.


𝔂 = 1 x² csc 2

2 x

Textbook Question

Slopes, Tangent Lines, and Normal Lines


In Exercises 31–40, verify that the given point is on the curve and find the lines that are (a) tangent and (b) normal to the curve at the given point.


x²y² = 9, (–1,3)